We present analytical results of the equation of state (EOS) described by a model of thermal quark mass and magnetic potential term introduced in Polyakov–Nambu–Jona–Lasinio (PNJL) model for two-flavour quarks. Under the influence of thermal quark mass and magnetic field term in the potential, the calculated results of EOS using the model are enhanced in a good pattern up to the temperature $T=2.2T_c$ MeV and can follow result similar to the unmagnetised field when the temperature is increased beyond $T=2.2T_c$ MeV. The result shows that the thermodynamic behaviour agrees well with the standard properties of quantum chromodynamics (QCD) thermodynamics and enhance the result up to 2.2$T_c$ from the earlier predicted results and show the same behaviour beyond 2.2$T_c$ MeV.

The recent research work of two-flavor quarks with magnetised Polyakov–Nambu–Jona–Lasinio (mPNJL) model has been upgraded with the inclusion of a finite chemical potential in the temperature-dependent quark mass and the chemical potential in the interaction potential of Lagrangian density in continuation of the work of Dahiya and Singh (Pramana – J. Phys. 94:120 (2020)). The thermodynamic quantity like equation of state (EOS) and its relevant thermodynamic properties after the inclusion of the finite chemical potential in the temperature-dependent quark mass and potential are represented in the results. The outputs from the calculations show that all the thermodynamic quantities are obtained in such a way that it is enhanced and improved from theearlier result of thermodynamic properties of zero chemical potential. It means that the chemical potential really plays the role of upgradation in the outputs of thermodynamic parameters in comparison to the output produced bythe absence of chemical potential in the thermal mass and potential of the Lagrangian density.